Abstract wave equations and associated Dirac-type operators

作者：Gesztesy Fritz; Goldstein Jerome A; Holden Helge^*; Teschl Gerald

来源：Annali di Matematica Pura ed Applicata, 2012, 191(4): 631-676.

DOI：10.1007/s10231-011-0200-7

• Dirac operators
• Supersymmetry
• Wave equations
• Semigroups
• Damping terms
• Quadratic operator pencils

We discuss the unitary equivalence of generators G (A,R) associated with abstract damped wave equations of the type u + R(u) over dot + A* Au = 0 in some Hilbert space H-1 and certain non-self-adjoint Dirac-type operators Q (A,R) (away from the nullspace of the latter) in H-1 circle plus H-2. The operator Q (A,R) represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of A*A. In addition to the unitary equivalence results concerning G (A,R) and Q (A,R) , we provide a detailed study of the domain of the generator G (A,R) , consider spectral properties of the underlying quadratic operator pencil , derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where R represents an appropriate function of |A| is treated in depth, and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R = 0) and damped (R not equal 0) abstract wave equations as well as the cases for some and (but 0 not an eigenvalue of A*A) are separately studied in detail.

• 出版日期2012-12